Marco Grandis

Topological cospans and their concatenation, by pushout, appear in the
theories of tangles, ribbons, cobordisms, etc. Various algebraic
invariants have been introduced for their study, which it would be
interesting to link with the standard tools of Algebraic Topology,
(co)homotopy and (co)homology functors.

Here we introduce collarable (and collared) cospans
between topological spaces. They generalise the cospans which appear in
the previous theories, as a consequence of a classical theorem on
manifolds with boundary. Their interest lies in the fact that their
concatenation is realised by means of homotopy pushouts. Therefore,
cohomotopy functors induce `functors' from collarable cospans to
spans of sets, providing - by linearisation - topological quantum field
theories (TQFT) on manifolds and their cobordisms. Similarly, (co)homology
and homotopy functors take collarable cospans to relations of abelian
groups or (co)spans of groups, yielding other `algebraic' invariants.

This is the second paper in a series devoted to the study of cospans in
Algebraic Topology. It is practically independent from the first, which
deals with higher cubical cospans in abstract categories. The third
article will proceed from both, studying cubical topological cospans and
their collared version.